They get really upset!
Take my home town: Washington lost not one but two baseball teams called the Washington Senators - the first one moved to become the Minnesota Twins, then a few years after scoring a team in the league expansion, the new Senators moved to become the (Dallas) Texas Rangers

(edit: just went back and read the Wiki on this... apparently, DC got that expansion team because the government threatened them with anti-trust action for letting Washington's team leave! I guess the threat eventually died down)
I was very young when it happened so I grew up in an environment of constant complaining about us not having a team. But people eventually sucked it up and supported the nearby Baltimore Orioles en masse. Eventually (and this happened since I moved abroad) we stole the nearly bankrupt Montreal Expos, who are now the Washington Nationals.
All quite messy. But at least we have a team

That's how most cities who end up on the receiving end view it. People line up to buy season tickets, while local governments pay for stadiums and infrastructure improvements. Anything to get a team either through league expansion or luring some hard luck team in a problematic town.
Another good example is the Cleveland Browns, one of the founding teams of the NFL. Their owner, claiming financial hardship, moved them to Baltimore (now called the Ravens after the Edgar Allen Poe poem - probably the coolest American football team name!). The uproar in the city was so loud that the league imposed terms upon which the team name, history, records, etc. stayed in Cleveland for whenever another ownership group wanted to start a new Browns, which happened a couple of seasons later.
You even have a small percentage of fans that will continue to support the team after the move.
But again, I like the idea of a team belonging to its locale. This moving around business is destructive.